White noise distribution theory
White noise distribution theory
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DOI：
10.1201/9780203733813
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发表时间：
1996
期刊：
影响因子：
--
通讯作者：
H. Kuo
中科院分区：
文献类型：
--
作者：
H. Kuo
白色噪音简介什么是白色噪音？一个简单的例子背景抽象维纳空间可数-Hilbert空间核空间Gel'fand三重白色噪声作为无穷维微积分的白色噪声空间Schwartz空间的重构检验和广义函数空间检验和广义函数的一些例子检验和广义函数的构造几种构造的一般思想从Hilbert空间和算子构造Kubo和Kubo的一般构造Kondratiev和Streit的Takenaka构造S变换Wick张量和多重Weiner积分S变换的定义广义函数的例子测试函数的连续形式和解析延拓测试函数的连续形式增长条件和范数估计测试函数的解析延拓Delta函数Donsker的Delta函数Kubo-Yokoi Delta函数Delta函数的连续性刻划定理广义函数的刻划广义函数的收敛性检验函数的刻划Wick积和卷积可积函数微分算子微分算子伴随算子乘法算子粗微分与梯度积分核算子启发式讨论积分核算子粗拉普拉斯与数算子λ算子平移算子表示定理傅里叶变换傅里叶变换表示的定义傅里叶变换的基本性质傅里叶变换的分解傅里叶-Fourier变换的Gauss变换刻划Fourier-Mehler变换初值问题的Laplacian算子半群的Gross Laplacian算子半群的Levy Laplacian算子的Levy Laplacian算子变换球面平均与Levy Laplacian的关系Gross与Levy Laplacian的关系沃尔泰拉Laplacian与Fourier变换的关系二维旋转白色噪声积分非形式动机Pettis和Bochner积分白色噪声积分Ito积分的扩展Ito公式的推广单侧白色噪声微分随机积分方程白色噪声积分方程费曼积分非形式推导白色噪声公式显式计算正广义函数正广义函数构造Lee表征希达测度附录A：附录B：其他公式参考书目符号索引列表
Introduction to White Noise What is White Noise? A Simple Example Background Abstract Wiener Spaces Countably-Hilbert Spaces Nuclear Spaces Gel'fand Triples White Noise as an Infinite Dimensional Calculus White Noise Space A Reconstruction of the Schwartz Space The Space of Test and Generalized Functions Some Examples of Test and Generalized Functions Constructions of Test and Generalized Functions General Ideas for Several Constructions Construction from a Hilbert Space and an Operator General Construction of Kubo and Takenaka Construction of Kondratiev and Streit The S-Transform Wick Tensors and Multiple Weiner Integrals Definition of the S-Transform Examples of Generalized Functions Continuous Versions and Analytic Extensions Continuous Versions of Test Functions Growth Condition and Norm Estimates Analytic Extensions of Test Functions Delta Functions Donsker's Delta Function Kubo-Yokoi Delta Function Continuity of the Delta Functions Characterization Theorems Characterization of Generalized Functions Convergence of Generalized Functions Characterization of Test Functions Wick Product and Convolution Integrable Functions Differential Operators Differential Operators Adjoint Operators Multiplication Operators Gross Differentiation and Gradient Integral Kernel Operators Heuristic Discussion Integral Kernel Operators Gross Laplacian and Number Operator Lambda Operator Translation Operators Representation Theorem Fourier Transforms Definition of the Fourier Transform Representations of the Fourier Transform Basic Properties Decomposition of the Fourier Transform Fourier-Gauss Transforms Characterization of the Fourier Transform Fourier-Mehler Transforms Initial Value Problems Laplacian Operators Semigroup for the Gross Laplacian Semigroup for the Number Operator Levy Laplacian Levy Laplacian by the S-Transform Spherical Mean and the Levy Laplacian Relationship Between Gross and Levy Laplacians Volterra Laplacian Relationship with the Fourier Transform Two-Dimensional Rotations White Noise Integration Informal Motivation Pettis and Bochner Integrals White Noise Integrals An Extension of the Ito Integral Generalization of Ito's Formula One-Sided White Noise Differentiation Stochastic Integral Equations White Noise Integral Equations Feynman Integrals Informal Derivation White Noise Formulation Explicit Calculation Positive Generalized Functions Positive Generalized Functions Construction of Lee Characterization of Hida Measures Appendix A: Notes and Comments Appendix B: Miscellaneous Formulas Bibliography List of Symbols Index